Properties

Label 6012.11
Modulus $6012$
Conductor $6012$
Order $498$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(6012)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([249,83,84]))
 
pari: [g,chi] = znchar(Mod(11,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(6012\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(498\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.ba

\(\chi_{6012}(11,\cdot)\) \(\chi_{6012}(47,\cdot)\) \(\chi_{6012}(191,\cdot)\) \(\chi_{6012}(203,\cdot)\) \(\chi_{6012}(239,\cdot)\) \(\chi_{6012}(263,\cdot)\) \(\chi_{6012}(275,\cdot)\) \(\chi_{6012}(299,\cdot)\) \(\chi_{6012}(311,\cdot)\) \(\chi_{6012}(383,\cdot)\) \(\chi_{6012}(419,\cdot)\) \(\chi_{6012}(455,\cdot)\) \(\chi_{6012}(491,\cdot)\) \(\chi_{6012}(515,\cdot)\) \(\chi_{6012}(551,\cdot)\) \(\chi_{6012}(563,\cdot)\) \(\chi_{6012}(599,\cdot)\) \(\chi_{6012}(623,\cdot)\) \(\chi_{6012}(671,\cdot)\) \(\chi_{6012}(695,\cdot)\) \(\chi_{6012}(731,\cdot)\) \(\chi_{6012}(743,\cdot)\) \(\chi_{6012}(767,\cdot)\) \(\chi_{6012}(815,\cdot)\) \(\chi_{6012}(839,\cdot)\) \(\chi_{6012}(851,\cdot)\) \(\chi_{6012}(911,\cdot)\) \(\chi_{6012}(923,\cdot)\) \(\chi_{6012}(947,\cdot)\) \(\chi_{6012}(959,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((3007,3341,4681)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{14}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{1}{498}\right)\)\(e\left(\frac{35}{498}\right)\)\(e\left(\frac{97}{249}\right)\)\(e\left(\frac{176}{249}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{47}{166}\right)\)\(e\left(\frac{8}{249}\right)\)\(e\left(\frac{1}{249}\right)\)\(e\left(\frac{233}{498}\right)\)\(e\left(\frac{7}{498}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{249})$
Fixed field: Number field defined by a degree 498 polynomial